Thermodynamics of Membrane Diffusion: Charged Ion - Online Tutor, Practice Problems & Exam Prep

In membrane diffusion for charged ions, the transmembrane potential (Δψ) is crucial, representing the difference in electrical charge across the membrane. The Gibbs free energy change (ΔG_transport) for ion transport is given by the equation: ${\mathrm{\Delta G}}^{\mathrm{transport}}=zf\mathrm{\Delta \psi }$, where z is the net charge of the ion and f is Faraday's constant. The electrochemical gradient combines both chemical and electrical gradients, essential for understanding ion diffusion across membranes.

Thermodynamics of Membrane Diffusion: Charged Ion

So now that we've covered the thermodynamics of membrane diffusion for uncharged molecules in our previous lesson videos, in this video, we're going to introduce the thermodynamics of membrane diffusion, but for charged ions. If we take a look at our image on the left-hand side, we'll see a reminder that in this video, we are specifically focusing on charged ions. Notice here in yellow what we have is our charged ion, more specifically our positively charged cation. In this video, we'll be focusing on the thermodynamics as charged ions like this one diffuse across a membrane through a membrane channel from their initial side to the final side of the membrane.

What's important to note is that unlike uncharged molecules, when charged ions diffuse across a membrane, the transmembrane potential or the transmembrane voltage, Δψ, must be considered. Recall from our previous lesson videos that we already introduced the transmembrane potential or the transmembrane voltage, Δψ, is really just equal to the difference in electrical charge across the membrane. If we consider only the transmembrane potential or the electrical gradient alone, then the ΔG transport of ions would be given by the following equation:

Δ G transport = z ∗ f ∗ Δ ψ

where z is equal to the net charge of the diffusing ion, and f is known as Faraday's constant, named after the scientist Michael Faraday. Faraday's constant f is equal to the magnitude of the charge of 1 mole of electrons and, despite electrons being negatively charged, f is always a positive value. In fact, f is equal to approximately 96,485, with units of joules per volts per moles or with units of coulombs per mole, depending on your textbook. It's important to note that when you have a number or a variable raised to the negative one, you need to take the reciprocal of it, giving us joules per volts per moles.

This equation would be true if only the transmembrane potential or electrical gradient alone was considered. However, remember from our previous lesson videos that when ions diffuse across a membrane, it does not only depend on the electrical gradient; it actually depends on the electrochemical gradient. The electrochemical gradient is a combination of the chemical gradient and the electrical gradient. If we look at our image on the right-hand side, notice that the blue shaded region here is the portion of the equation explained by the chemical gradient. This blue portion alone is the same equation we introduced in our previous lesson videos when we talked about the thermodynamics of membrane diffusion for uncharged molecules, which do not depend on electrical gradients, only on chemical gradients. Because ions depend on the electrochemical gradient, they require both the chemical gradient portion and the addition of the electrical gradient portion of the equation:

z ∗ f ∗ Δ ψ

When considering the thermodynamics of membrane diffusion for charged ions, all you need to do is add this electrical gradient portion to the equation, and then follow the same steps as before to find your answer. This concludes our lesson, an introduction to the thermodynamics of membrane diffusion for charged ions. In our next video, we will show you an example of how to apply this entire equation to calculate the thermodynamics of membrane diffusion for charged ions. I'll see you in that next video.

Thermodynamics of Membrane Diffusion: Charged Ion Example 1

Alright. So here we have an example problem that wants us to calculate the energy cost or the delta G transport of pumping calcium ions or Ca²⁺ from the cytosol to the extracellular space. If the temperature is equal to 37 degrees Celsius, the transmembrane potential delta psi is equal to 0.05 volts where the inside of the cell is still negative. The cytosolic calcium concentration is equal to 1.0 × 10^-7 molar and the extracellular calcium concentration equal to 1.0 × 10^-3 molar. And so, in order to solve problems like this one, we're actually going to follow a four-step process that's very similar to the four-step process that we covered in our previous lesson videos with just a little bit of differences.

Now in step number 1, it's the same as step number 1 from our previous lesson videos. So all we need to do is determine the net charge of the diffusing molecule. And if we take a look at our example problem, of course, the diffusing molecule is calcium because that's what's being pumped across the membrane. And calcium here is clearly a charged ion, so we can go ahead and check off this box. And, of course, by the 2 plus here, we know that its charge is actually plus 2 and so we can write that here in this blank. And so because this is a charged ion, of course, what this means is that we are going to use the same equation that we introduced from our previous lesson video that includes both the chemical gradient and the electrical gradient since again charged ions diffuse based on the electrochemical gradient, the combination of both of these.

So moving on to step number 2, which is actually slightly different from step number 2 in our previous lesson videos. And here, we're still going to determine the direction of diffusion by establishing the initial and the final sides. But we're also going to determine the sign of the transmembrane potential delta psi whether or not it's going to be positive or negative when we include it into our equation. Drawing a sketch is sometimes going to be very helpful for step number 2. And so notice down below we have our cell membrane and you can see that we have the ion channel embedded. And so, of course, to determine the direction of diffusion, looking back at our example problem, notice that we're pumping calcium from the cytosol to the extracellular space. Since the cytosol is the initial side and the extracellular space the final side, delta psi will be calculated as psi (final) - psi (initial). Due to the inner cell being more negative relative to the outer cell which is more positive, delta psi results in a positive value. Thus in our equations, we must ensure that delta psi retains this positive value.

Regarding step number 3, it remains the same as in our previous lessons. We need to check all the units involved such as converting the temperature to Kelvin by adding 273.15 to 37 degrees Celsius, resulting in 310.15 Kelvin. The transmembrane potential given already matches the unit requirements for our equations in volts, and the concentrations provided are adequate as they are in molarity, which cancel each other out in a ratio form during calculations.

Finally, in step number 4, we plug all our values into the appropriate equations previously defined. The 'R' value used is 8.315 joules per mole-Kelvin, and Faraday's constant 'F' is 96,485 joules per volt-mole. After doing the algebraic calculations, delta G transport turns out to be 33,400.917 joules per mole. To convert this to kilojoules per mole for easier comparison, we divide the resultant value by 1000, resulting in about 33.4 kilojoules per mole. This concludes the detailed explanation of the delta G transport of pumping calcium under these conditions.

Moving forward in our course, we're going to get some practice applying each of these four steps to solve problems similar to this one. This concludes this example problem, and I'll see you guys in our next video.

Calculate the free energy change (ΔG _transport) for the movement of Na ⁺ into a cell when its concentration outside is 150 mM and its cytosolic concentration is 10 mM. Assume that T = 20°C and ΔΨ = –50 mV (inside negative).

• A. –1.7 KJ/mol.
• B. –11.4 KJ/mol.
• C. –11,600 KJ/mol.
• D. 11.4 KJ/mol.
• E. 14.3 KJ/mol.

Calculate the ΔG_transport required to move 1 mole of Na ⁺ ions from inside the cell ([Na⁺] inside = 5 mM) to the outside of the cell ([Na ⁺] outside = 150 mM) when ΔΨ = –70 mV (inside negative) & the temperature is 37°C.

• A. –15.5 KJ/mol.
• B. 2.0 KJ/mol
• C. 15.5 KJ/mol.
• D. –2.0 KJ/mol.

Calculate the ΔG_transport when Ca²⁺ ions move from the endoplasmic reticulum ([Ca²⁺] = 1 mM) to the cytoplasm ([Ca²⁺] = 0.1 μM). Assume that ΔΨ = 0 and T = 25°C.

• A. 23 KJ/mol.
• B. –23 KJ/mol.
• C. –17 KJ/mol.
• D. 17 KJ/mol.

Here’s what students ask on this topic:

What is the role of the transmembrane potential (Δψ) in the diffusion of charged ions across a membrane?

The transmembrane potential (Δψ) represents the difference in electrical charge across a membrane and is crucial for the diffusion of charged ions. Unlike uncharged molecules, charged ions are influenced by both the chemical gradient and the electrical gradient. The transmembrane potential contributes to the electrochemical gradient, which determines the direction and rate of ion diffusion. The Gibbs free energy change (ΔG_transport) for ion transport is given by the equation:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ

where z is the net charge of the ion and f is Faraday's constant. This equation shows how the transmembrane potential directly affects the energy required for ion transport.

How does Faraday's constant (f) factor into the thermodynamics of membrane diffusion for charged ions?

Faraday's constant (f) is a key factor in the thermodynamics of membrane diffusion for charged ions. It represents the magnitude of the charge of one mole of electrons and is always a positive value, approximately 96,485 coulombs per mole. In the context of ion transport, Faraday's constant is used in the equation for the Gibbs free energy change (ΔG_transport):

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ

Here, z is the net charge of the ion and Δψ is the transmembrane potential. Faraday's constant helps quantify the energy change associated with moving charged ions across a membrane, making it essential for understanding the electrochemical gradient.

What is the difference between the chemical gradient and the electrical gradient in membrane diffusion?

The chemical gradient and the electrical gradient are two components of the electrochemical gradient that influence membrane diffusion. The chemical gradient refers to the concentration difference of a substance across a membrane, driving diffusion from high to low concentration. The electrical gradient, on the other hand, is the difference in electrical charge across the membrane, influencing the movement of charged ions. For charged ions, both gradients must be considered together, forming the electrochemical gradient. The Gibbs free energy change (ΔG_transport) for ion transport combines these gradients:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ+RTln(C_final/C_initial)

where z is the ion's charge, f is Faraday's constant, Δψ is the transmembrane potential, R is the gas constant, T is the temperature, and C represents concentrations.

How do you calculate the Gibbs free energy change (ΔG_transport) for the diffusion of charged ions across a membrane?

To calculate the Gibbs free energy change (ΔG_transport) for the diffusion of charged ions across a membrane, you need to consider both the chemical and electrical gradients. The equation is:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ+RTln(C_final/C_initial)

where z is the net charge of the ion, f is Faraday's constant (96,485 C/mol), Δψ is the transmembrane potential, R is the gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin, and C represents the ion concentrations on either side of the membrane. This equation combines the effects of the electrical gradient (z·f·Δψ) and the chemical gradient (RT ln(C_final/C_initial)) to determine the total energy change for ion transport.

Why is the electrochemical gradient important for the diffusion of charged ions across membranes?

The electrochemical gradient is crucial for the diffusion of charged ions across membranes because it combines both the chemical and electrical gradients. The chemical gradient is the concentration difference of ions across the membrane, while the electrical gradient is the difference in electrical charge. Together, they determine the direction and rate of ion movement. The Gibbs free energy change (ΔG_transport) for ion transport is given by:

$\mathrm{\Delta G}$_transport=z⋅f⋅Δψ+RTln(C_final/C_initial)

where z is the ion's charge, f is Faraday's constant, Δψ is the transmembrane potential, R is the gas constant, T is the temperature, and C represents concentrations. This equation shows how both gradients influence ion transport, making the electrochemical gradient essential for understanding membrane diffusion.